Multi-Dimensional Uncertain Calculus with Liu Process

Abstract Uncertain calculus deals with the integral and differential of some uncertain processes. So far, uncertain integrals have been defined with respect to Liu process, renewal process, finite variation process, and multiple Liu processes. This paper presents an uncertain integral of a matrix of uncertain processes with respect to multi-dimensional Liu process, and verifies its linearity property. Then an uncertain differential of a multi-dimensional uncertain process with respect to a multi-dimensional Liu process is defined, and a fundamental theorem is derived. In addition, a concept of multi-dimensional uncertain differential equation is proposed, and solutions of some special types of multi-dimensional uncertain differential equations are given